Existence of additive uniqueness sets with prescribed minimal number of summands

Let $n$ be an integer satisfying $n\le 2$. Existence conjecture for additive uniqueness sets. There exist $E\subseteq\mathbb{N}$ and a set $S$ of arithmetic functions such that $E$ is an $n$-additive uniqueness set for $S$ but is not a $k$-additive uniqueness set for $S$ when $k<n$. The source gives no resolution or further qualification of this assertion.